TITLE: Discrete Laplacian
QUESTION [5 upvotes]: I was wondering if anybody would explain (or please point me to a nice reference) as to why the ``discrete Laplacian" on a graph is actually called a Laplacian. Namely, how is it related to the standard Laplacian on $\mathbb{R}^n$ ? Is there a sense in which the discrete Laplacian on say, a square lattice, would converge to the standard Laplacian as the lattice spacing tends to $0$ ?

REPLY [5 votes]: I like the discussion in section 5.6 of Gilbert Strang's book Differential Equations and Linear Algebra.  For a directed graph, the incidence matrix $A$ is a difference matrix --- so it is a discrete analog of the gradient $\nabla$.  The graph Laplacian is $A^T A$, which is analogous to the (negative) Laplacian $\nabla^T \nabla = -\text{div} \nabla$.